In September I made my second visit to Sicily, spending 22 days traveling counterclockwise around the island, starting and finishing in Catania, mostly using public transportation to get from place to place. I spent lots of time researching the forum posts about Sicily and got lots of suggestions and good ideas from many of them. On my trip I visited the well known tourist destinations of Siracusa, Catania, Taormina, Etna, Palermo, Agrigento, etc but also purposely stayed at or visited smaller towns not always on the tourist track. Here are some ideas for places not always considered.
One of my favorites was Mazara del Vallo, a small fishing town with heavy Tunisian influence, on the southwest corner of Sicily. Becky, a contributor to this forum had suggested I view the documentary titled Sicily Unpacked on youtube which follows the travels of an Italian chef and a British art historian as they eat and pursue their interests around the island. The documentary is wonderful and will give you lots of good ideas. Mazara del Vallo was one of the towns visited and it caught my interest. We took a train from Palermo to get there and stayed two nights, which was just the right amount of time. We spent one day exploring the town and seeing the Dancing Satyr that was pulled from the sea, and the second day we took a short train ride over to Castelvetrano and right in front of the train station we caught a public bus for a short ride to the beautiful ruins at Selinunte. I very much liked exploring Selinunte, right on the ocean, with far fewer people than Agrigento. i also recommend a visit to Erice. You will have a car so Erice is doable. The town was enveloped in fog and a light drizzle when we were there but still quite beautiful. If you were to drive from Palermo you could combine Erice with a visit to Trapani or the nature reserve in Scopello or the beach at San Vito lo Capo and then on to Mazara del Vallo and From Mazara del Vallo you can head over to Agrigento.
If you have any interest in religious sites, you might consider a visit to Tindari where the Sanctuary of the Black Madonna sits on a cliff overlooking the most beautiful sea views. There are also Greek and Roman ruins right there, walkable from the church. We stayed one night in the tiny beach town of Oliveri at the base of the cliff. But with a car I would visit Tindari church and ruins for a few hours and then continue on my way. Tindari is on the northern coast, not far from Milazzo. Milazzo, has a castle and is on the sea, It also caught my eye but I did not get there. From Tindari you could drive to Cefalu with possible stops in Castelbuono or Caccamo, mountain towns with castles and views. Cefalu is a very pleasant town to visit, and the historical center is very compact.
If you are a fan of the film “Cinema Paradiso” look into the small interior mountain town of Palazzo Adriano, where some of movie was filmed. I did not end up going there because of transportation, but it looks so very picturesque that it caught my eye and had I had a car I would have gone.